Abstract

A kink in the electronic dispersion associated with the active optical phonon, an anomalous change in the phonon spectrum at the superconducting transition, and strong isotope effects indicate that phonons play an interesting role in the cuprates. This opens the question of how phonon-mediated mechanisms for superconductivity could fit in with the claims of d-wave order parameters. Since the magnitude of the electron-phonon coupling, and the energy of the phonon mode in cuprates are outside the limited region of applicability for BCS theory, more sophisticated schemes need to be developed. I describe an approach that extends the Eliashberg theory through systematic expansion in the vertex function. By examining limiting behaviours of the electron-phonon problem for optical phonon modes, including the mapping to the BCS Hamiltonian, I argue why vertex corrections are essential for examining pairing with angular momentum, even for weak coupling. An extended scheme for the superconducting state leads to the important conclusion that d-wave superconductivity can be mediated by phonons, with the inclusion of Coulomb repulsion stabilising d-pairing.